# American Institute of Mathematical Sciences

2015, 2015(special): 1034-1040. doi: 10.3934/proc.2015.1034

## Recurrence of multi-dimensional diffusion processes in Brownian environments

 1 Colledge of Science and Technology, Nihon University, Narashino-dai, Funabashi 274-8501, Japan 2 Faculty of Science and Technology, Keio University, Hiyoshi, Yokohama 223-8522, Japan

Received  September 2014 Revised  February 2015 Published  November 2015

We consider limiting behavior of multi-dimensional diffusion processes in two types of Brownian environments. One is given values at different $d$ points of a one-dimensional Brownian motion, which is supposed to be a multi-parameter environment, and other is given by $d$ independent one-dimensional Brownian motions. We show recurrence of multi-dimensional diffusion processes in both Brownian environments above for any dimension and almost all environments. Their limiting behavior is quite different from that of ordinary multi-dimensional Brownian motion. We also consider cases of reflected Brownian environments.
Citation: Hiroshi Takahashi, Yozo Tamura. Recurrence of multi-dimensional diffusion processes in Brownian environments. Conference Publications, 2015, 2015 (special) : 1034-1040. doi: 10.3934/proc.2015.1034
##### References:
 [1] T. Brox, A one-dimensional diffusion process in a Wiener medium. Ann. Probab., 14 (1986), 1206-1218. [2] M. Fukushima, S. Nakao and M. Takeda, On Dirichlet form with random date - recurrence and homogenization, in "Stochastic Processes - Mathematics and Physics II (Bielefeld, 1985)'' (eds. S. Albeverio, Ph. Blanchard and L. Streit), Lect. Notes in Math., 1250, Springer-Verlag, (1987), 87-97. [3] K. Ichihara, Some global properties of symmetric diffusion processes. Publ. RIMS, Kyoto Univ., 14 (1978), 441-486. [4] K. Itô, On the ergodicity of a certain stationary process. Proc. Imp. Acad., 20 (1944), 54-55. [5] K. Itô and H.P. McKean,Jr., "Diffusion Processes and Their Sample Paths,'' Springer-Verlag, New York, 1965. [6] D. Kim, Some limit theorems related to multi-dimensional diffusions in random environments, J. Korean Math. Soc., 48 (2011), 147-158. [7] G. Maruyama, The harmonic analysis of stationary stochastic processes, Mem. Fac. Sci. Kyushu Univ., A4 (1949), 45-106. [8] P. Mathieu, Zero white noise limit through Dirichlet forms, with application to diffusions in a random environment, Probab. Theory Related Fields, 99 (1994), 549-580. [9] Y. Sinai, The limit behavior of a one-dimensional random walk in a random environment, Theory Probab. Appl., 27 (1982), 256-268. [10] F. Solomon, Random walks in a random environment, Ann. Probab., 3 (1975), 1-31. [11] H. Takahashi, Recurrence and transience of multi-dimensional diffusion processes in reflected Brownian environments. Statist. Proba. Lett., 69 (2004), 171-174. [12] H. Tanaka, Limit distributions for one-dimensional diffusion process in self-similar random environments, in "Hydrodynamic Behavior and Interacting Particle Systems (Minneapolis, Minn., 1986)'' (ed. G. Papanicolau), IMA Vol. Math. Appl. 9. Springer, (1987), 189-210. [13] H. Tanaka, Recurrence of a diffusion process in a multi-dimensional Brownian environment, Proc. Japan Acad. Ser. A Math. Sci., 69 (1993), 377-381.

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##### References:
 [1] T. Brox, A one-dimensional diffusion process in a Wiener medium. Ann. Probab., 14 (1986), 1206-1218. [2] M. Fukushima, S. Nakao and M. Takeda, On Dirichlet form with random date - recurrence and homogenization, in "Stochastic Processes - Mathematics and Physics II (Bielefeld, 1985)'' (eds. S. Albeverio, Ph. Blanchard and L. Streit), Lect. Notes in Math., 1250, Springer-Verlag, (1987), 87-97. [3] K. Ichihara, Some global properties of symmetric diffusion processes. Publ. RIMS, Kyoto Univ., 14 (1978), 441-486. [4] K. Itô, On the ergodicity of a certain stationary process. Proc. Imp. Acad., 20 (1944), 54-55. [5] K. Itô and H.P. McKean,Jr., "Diffusion Processes and Their Sample Paths,'' Springer-Verlag, New York, 1965. [6] D. Kim, Some limit theorems related to multi-dimensional diffusions in random environments, J. Korean Math. Soc., 48 (2011), 147-158. [7] G. Maruyama, The harmonic analysis of stationary stochastic processes, Mem. Fac. Sci. Kyushu Univ., A4 (1949), 45-106. [8] P. Mathieu, Zero white noise limit through Dirichlet forms, with application to diffusions in a random environment, Probab. Theory Related Fields, 99 (1994), 549-580. [9] Y. Sinai, The limit behavior of a one-dimensional random walk in a random environment, Theory Probab. Appl., 27 (1982), 256-268. [10] F. Solomon, Random walks in a random environment, Ann. Probab., 3 (1975), 1-31. [11] H. Takahashi, Recurrence and transience of multi-dimensional diffusion processes in reflected Brownian environments. Statist. Proba. Lett., 69 (2004), 171-174. [12] H. Tanaka, Limit distributions for one-dimensional diffusion process in self-similar random environments, in "Hydrodynamic Behavior and Interacting Particle Systems (Minneapolis, Minn., 1986)'' (ed. G. Papanicolau), IMA Vol. Math. Appl. 9. Springer, (1987), 189-210. [13] H. Tanaka, Recurrence of a diffusion process in a multi-dimensional Brownian environment, Proc. Japan Acad. Ser. A Math. Sci., 69 (1993), 377-381.
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